Question

Find the area of the regular 18 gon with a radius of 13mm

Answer 1

This polygon thing is all about triangles. If the polygon has 18 sides, it also has 18 central angles.  360/18  = 20.  This means that each one of the triangles has a vertex angle of 20.  We need to take this one triangle now and "pull" it out and isolate it cuz we only need to worry about 1, and we need to find the base measure of it for the perimeter, and the altitude of it for the apothem.  It would be easiest then to separate this triangle in half and have it be a right triangle, and the vertex angle is now 10.  If the vertex angle is 10 and this is a right triangle, then the 3rd angle by the triangle angle-sum theorem has to be 80.  The side ACROSS from the 80 degree angle is the apothem and the info telling us that the radius is 13 now puts that 13 as the hypotenuse of our right triangle.  We will use a trig identity relating this reference angle of 80 to the side across from it, "a", and the hypotenuse of 13.  $sin(80)= \frac{x}{13}$ and we get that that side measures 12.8, rounded.  Now we have the apothem for the formula.  Now we need the base measure of the right triangle.  Use cos of 80 to find it: $cos(80)= \frac{x}{13}$ ad we get the base measure of the right triangle is 2.25.  But don't forget that this is only half the original triangle we pulled out, so the base of the "whole" triangle is 2.25 * 2 which is 4.5.  If one side measures 4.5 and we have 18 of them, the perimeter for our formula is 81.  Now let's put everything together in the formula to find the area for the polygon: $A= \frac{1}{2} (12.8)(81)$ and we find that the area is 518.4